Help a math major choose his classes

[numb3rspunk]

For my undergraduate mathematics program the required classes are Calculus 1,2,3, Linear Algebra, Differential Equations, Introduction to Advanced Mathematics, Advanced Calculus, Algebraic Structures, and Mathematical Statistics I. Then I am told to choose 3 classes from each of the lists below. I am planning to go to graduate school for math and i was wondering which 3 classes from each list would benefit me in the future. I have underlined the ones that seem important in my opinion but i would like to know others peoples opinions. Which ones are not useful at all. Which ones are the most important.. and so on.

List 1
Introduction to Combinatorics
Complex Variables
College Geometry
Number Theory
Topics in Advanced Calculus
Topics in Algebraic Structures
Topology

List 2
Advanced Differential Equations
Graph Theory
Mathematical Modeling
Mathematical Statistics II
Numerical Analysis
Mathematical Logic
Axiomatic Set Theory

 

[roam]

Well, it 100% depends on your major. Are you majoring in Pure Mathematics or in Applied Mathematics?

 

[ekrim]

That's not a bad list there, well rounded and all. But, yea like roam said, it depends on where you're heading.

 

[ralphhumacho]

Take as many as possible!

If you're going to study pure math in grad school, Number Theory, Topology, Complex, and Topics in Alg Structures (which I assume is Advanced Abstract Algebra) are MUST HAVES.

If your going the applied route, your course selection will depend heavily on what you want to specialize in during grad school (stats vs. analysis etc...) . However, definitely take as much Real Analysis as you can.

Hope this helps.

 

[zhentil]

If you're going into grad school in math, I would take complex, topology, and topics in advanced calculus (probably will cover undergrad analysis, no?). You will need to take these in grad school, and they will be tested on the GRE.

As far as the other ones go, it's up to your personal taste. You can be a very successful researcher in mathematics and never study logic, combinatorics, graph theory, statistics, etc. at all.

Numerical analysis, advanced differential equations, mathematical modeling, and statistics will be useful only if you plan to go into applied math or industry.

As an aside, it seems pretty bizarre to have two undergrad classes in logic and none in, say, differential geometry. Does your department have a lot of logicians or something?

 

[numb3rspunk]

Well my school doesn't offer the choice between pure or applied mathematics, so I am simply majoring in mathematics. I don't really know which i would want to go into (pure or applied) because they both seem to have their pros and cons.

 

[numb3rspunk]

so what your saying is if after taking the classes if i prefer advanced differential equations, numerical analysis, and mathematical statistics i should go for applied and if i prefer advanced calculus, abstract algebra, and complex analysis i should go for pure?

 

[will.c]

I know you said you underlined the ones that seem the most important for grad school (although as everyone has noted, the importance is relative depending on whether you want to go into pure or applied mathematics)

Here's an exercise: Read the course descriptions for each class, then come back and tell us which ones sound the most interesting to you - if there was nothing else that went into the decision, which classes of the sets do you think you would most enjoy?

 

[jtbell]

How far along are you now in your program? If you're in the early stages (beginning of freshman or sophomore year), you probably haven't been exposed to enough math yet to have a good sense of which way you'd like to end up going.

 

[numb3rspunk]

well i sort of did a backwards route and decided first which classes i thought were unnecessary (which I am not sure about). i figured no math major takes college geometry unless their majoring in math education. i didnt think the logic classes (mathematical logic and axiomatic set theory) sound neccesary nor fun (but i may be wrong). graph theory doesn't seem useful for graduate classes. mathematical modeling seems fun but also seems as its not very necessary.

this leaves me with the 3 underlined classes in list 2 which all seem more useful than the rest of the ones in list 2. and 6 classes in list 1 (all excluding college geometry) of which i have to pick 3. i just want to know which 3 are the most important and if you can't choose only 3.. choose more (meaning i would have to take extra classes). and do u agree with my underlined in list 2.

i am entering my junior year

 

[hubris]

Junior year, strikes me as odd that you do not have a taste of applied/pure maths.
I would encourage you to take another look at the math modeling course, as a fall back at the very least. I imagine you will get exposure to several different problems and some coding exposure.

 

[numb3rspunk]

well i have taken calc 1,2,3, linear algebra, and diff eq.

my junior year will be:
fall- intro to adv math
math stats 1
*course from math list*
spring- adv calc
algebraic structures
*course from math list*

senior year
fall- *course from math list*
*course from math list*
spring- *course from math list*
*course from math list*

and i was hoping you guys would help me choose those 6 from the list (3 from each)

i will be getting a taste of pure math in the fall. unless linear algebra counts as pure and diff eq counts as applied. then i would definitely have to say i enjoyed diff eq a lot more than linear algebra

 

[PowerIso]

graph theory doesn't seem useful for graduate classes.

Are you kidding me? Graph theory rocks!

Although you'll get your fair dose of it in "Introduction to Combinatorics."

Also, I think Topology should be in every math majors education.

 

[will.c]

I agree that you should absolutely take topology, and I also think you should take algebraic structures. Intro to combinatorics? Without the course description I can't tell for sure what you cover, but from personal preference, I say you skip it (but take graph theory from set 2). I'd recommend complex variables as the third, mostly because it's fun.

Filling out set 2, I think advanced diff.eq.,... and I guess numerical analysis, even though it's mind numbingly boring.

Granted, this is simply what I would do if taking undergrad classes over again... And assuming I was a math major.

 

[eastside00_99]

I would say avoid doing things like complex variables, topics in analysis, and topology or say combinatorics, number theory, topics in algebraic structures. If you are preparing for graduate school, I would recommend topics in algebraic structures and topics in analysis. Then I would say you should consider the other one from list 1 as an elective, but I would probably not choose combinatorics or college geometry. I think topology would be the most useful.

From list 2, well gosh, I took graph theory and that counted as a science elective not a math elective --- the rest of them I never even looked at. I always considered myself as someone going to grad school for pure math. If it were me, I would take math logic, axiomatic set theory, and graph theory...but its not and you may have an interest in applied math. I think if you are not sure, you should definitely take mathematical modeling. Then I would say graph theory and differential equations.

This is advice if you are not sure on what area of math you want to study. It would definitely change if you said 'I want to do pure math' or 'I want to do applied'.

 

[quasar987]

In my opinion, the one thing that really matters if you're planning on going to grad school (applied or pure) is that you take topology.

For the rest, go with your personal taste, but imho, Complex Variables is the second most useful.

 

[numb3rspunk]

which classes do you guys think arent necessary?

 

[SCV]

which classes do you guys think arent necessary?

All of those can be necessary depending on what part of math you are interested. Since you have no real preference we can't tell you what isn't necessary. Later you may figure out that you like something for which a class we said was not necessary actually is.

 

[zhentil]

which classes do you guys think arent necessary?

It really depends what you want to go into. That's why a lot of people have been advocating taking classes that will server you no matter what you go into. No matter what, if you go to grad school in math, you will need to be a wizard at algebra and analysis (if only to pass quals). Same can be said to a lesser degree about complex, since some schools require a qual in it, and also because it pops up in almost every area of math. Undergrad topology might be nice, if only to make metric space analysis easier.

After that, it completely depends on the field you want to go into. If you want to go into something related to algebra, topology, etc. you will find almost every other class on the list useless. If you want to go into something applied, you would probably benefit from advanced differential equations, numerical analysis, and math modeling.

But once you have the courses you think will prepare you for grad school, just take the courses that sound fun/interesting. If you think graph theory sounds cool, take it. If you think Mathematical Statistics II sounds interesting, don't go to grad school in math (kidding).

 

[joelio36]

Complex Variables
Number Theory
Topics in Advanced Calculus

Advanced Differential Equations
Graph Theory
Mathmatical Logic

I would choose these as they will provide the most solid mathematical foundations.
Number Theory and Graph Theory- If you want to discover anything, a must.
Complex Variables and Mathematical Logic- Great for when writing your dissertation

 

[hubris]

Well, if you want a list:

Section 1:
Complex Variables
Topology
Topics in advanced Calculus

Section 2:
Advanced differential equations (or math stat II if you like I)
Mathematical modeling
Numerical Analysis

 

[zpconn]

Topology is a must. Personally I don't know how a person could call himself a mathematician if he didn't know some topology. Topology is for mathematics what evolution is for biology--it's that basic, unifying scheme that brings everything together.

I think it's also important to be good with complex analysis. That subject seems to pop up a lot in random places (number theory--the last place you'd expect to find it!) and it's simply beautiful.

Other than that, it's really up to you. You should probably take one or the other of Topics in Advanced Calculus and Topics in Algebraic Structure. Do you like analysis or algebra more? :D

List 2 is completely subjective in my opinion. Differential equations are useful in all different fields of mathematics, so I'd strongly recommend that. As for the others, pick what you like. None of the others are at the same level of significance as topology, complex analysis, advanced calculus/algebra, and differential equations. They're more of "niche" subject areas.

 

[roam]

Hi@Numb3rspunk

These are the courses you have to study for each major, in my university.


A major in applied mathematics;

Modelling and Computation
Calculus & Linear Algebra I
Calculus & Linear Algebra II
Calculus & Linear Algebra III
Differential Equations
Numerical Computation
Advanced Modelling and Computation
Partial Differential Equations
Real and Complex Calculus
Methods in Applied Mathematics


A major in pure mathematics;

Calculus & Linear Algebra I
Calculus & Linear Algebra II
Calculus & Linear Algebra III
Principles of Mathematics
Combinatorial Computing
Mathematical Logic
Algebra and Applications
Algebraic Structures
Geometry and Topology
Multivariable Calculus
Real Analysis
Analysis in Higher Dimensions

(There are, of course, optional courses in pure math, such as; "history of mathematics" and "mathematics education".)

A degree with a focus on Pure Mathematics is an excellent qualification for a career in teaching or research.
Whereas applied math is about mathematical techniques that are used in other fields such as Engineering.
It is the job of an applied mathematician to show how mathematical techniques can be applied to science and technology to answer interesting questions.

But you can study pure if you see intrinsic beauty and usefulness within this subject.

 

[adartsesirhc]

I agree with just about everybody here that Topology and Complex Analysis are absolutely necessary. I would take Advanced Calculus and Advanced Differential Equations, too, just because of their usefulness and importance in math.

 

[roam]

You need to study advanced calculus whether you’re majoring in applied or pure -- it doesn't matter. Differential equations is something applied mathematicians focus on a lot.

Later on you'll see many topics do in fact share certain fundamental underpinnings. But I can't really help you too much because the syllabus here is different from the one in USA.

 

[numb3rspunk]

is number theory necessary if I am already going to be taking algebraic structures?

and is it safe to say that if i like numerical analysis then i would like mathematical modeling?

 

[zpconn]

is number theory necessary if I am already going to be taking algebraic structures?

That probably depends on exactly what's covered in the number theory class. I've seen some classes that are dual introductions to number theory and abstract algebra. Other number theory classes have abstract algebra as a prerequisite, and still others are at a more basic level and avoid doing much in the way of abstract algebra.

 

[numb3rspunk]

MAS 4213 Number Theory: Topics to be discussed are selected from the following: congruences, Diophantine equations, distribution of primes, primitive roots, quadratic reciprocity, and classical theorems of number theory. Prerequisite is Calc 2

MAS 4301 Algebraic Structures: An introduction to abstract mathematical structures of modern algebra. Fundamental concepts of groups, rings, and fields will be studied. Prerequisite is Intro to Adv Math

 

[mathwonk]

abstract algebra is a study of useful fundamental concepts that occur everywhere. number theory is a specialized study of specific examples that gave rise to many ideas of abstract algebra, but are of interest mainly for themselves. number theory is relatively narrow and special, abstract algebra is quite broad and general.

however the interest and depth of the examples in number theory led to the discovery of many of the general ides in algebra. so it may help to study both.